UNIVERSITY OF NEW BRUNSWICK

DEPARTMENT OF MATHEMATICS & STATISTICS
MATH 2213

3 Hours
December 1998
FINAL EXAM

MARKS
(7)1. Let ${\displaystyle \; A = \left( \begin{array}{ccr}
1 & 3 & 8\\
1 & 2 & 5\\
2 & 3 & 10
\end{array} \right)}$. Find A-1 and use it to solve ${\displaystyle \; AX = \left( \begin{array}{r}
-3\\ 1\\ 3
\end{array} \right)}$ and ${\displaystyle \; AY = \left(
\begin{array}{c}
5\\ 5\\ 4
\end{array} \right)}$.

(8)2. \begin{alist}\item Find \marginpar{(8)} an LU-factorization for ${\displaystyle ...
...\begin{array}{c}
2\\ 0\\ 3
\end{array} \right) \; .}$\\ [0.25in]
\par\end{alist}

(6)3. Find bases for col A, row A and nul A if ${\displaystyle
\;\;\; A = \left( \begin{array}{rrrr}
1 & -3 & 2 & 5\\
-2 & 6 & 0 & -3\\
4 & -12 & -4 & -1
\end{array} \right) \;}$.


(5)4. \begin{alist}\item Show \marginpar{(5)} that $k = 1$\space is an eigenvalue for ...
... 0 & 1 & 2\\
0 & 0 & 2 & 4
\end{array}\right)\; }$ .\\ [0.25in]
\par\end{alist}

5. Let ${\displaystyle \; B = \left\{ \left( \frac{6}{7}, \;
\frac{2}{7}, \; \frac{-3}{...
..., \; \left( \frac{3}{7}, \;
\frac{-6}{7}, \; \frac{2}{7} \right) \right\} \; .}$

(5)
\begin{alist}\item Show \marginpar{(5)} $B$\space is an orthonormal set.
\item F...
... the change of coordinates matrix from $S$\space to $B$ .\\ [0.10in]
\end{alist}

(3)6. \begin{alist}\item Show \marginpar{(3)} that the system
\par\begin{tabular}{rrrr...
...npar{(7)} the least squares solution(s) to the above.\\ [0.10in]
\par\end{alist}

(7)7. Suppose ${\displaystyle T(x) = Ax \;\; \mbox{where} \;\; A =
\left( \begin{array}{cc}
1 & 2\\
2 & 1
\end{array} \right) \; .}$


\begin{alist}\item Give a basis for which $[T]_B$\space is diagonal and give $[T...
... .
\item Use (b) to find at least one entry in $\; A^4$ .\\ [0.10in]
\end{alist}

(4)8. \begin{alist}\item Let \marginpar{(4)} $v_0$\space be a fixed vector in $\Bbb R^...
...s the rank of $A$ , and what
does this tell you about $A \;$\space ?
\end{alist}